This thesis details a Python-based software designed to calculate the Jones polynomial, a vital mathematical tool from Knot Theory used for characterizing the topological and geometrical complexity of curves in \( \mathbb{R}^3 \), which is essential in understanding physical systems of filaments, including the behavior of polymers and biopolymers. The Jones polynomial serves as a topological invariant capable of distinguishing between different knot structures. This capability is fundamental to characterizing the architecture of molecular chains, such as proteins and DNA. Traditional computational methods for deriving the Jones polynomial have been limited by closure-schemes and high execution costs, which can be impractical for complex structures like those that appear in real life. This software implements methods that significantly reduce calculation times, allowing for more efficient and practical applications in the study of biological polymers. It utilizes a divide-and-conquer approach combined with parallel computing and applies recursive Reidemeister moves to optimize the computation, transitioning from an exponential to a near-linear runtime for specific configurations. This thesis provides an overview of the software's functions, detailed performance evaluations using protein structures as test cases, and a discussion of the implications for future research and potential algorithmic improvements.

翻译：本论文详细介绍了一款基于Python的琼斯多项式计算软件。琼斯多项式是纽结理论中用于刻画 \( \mathbb{R}^3 \) 空间中曲线拓扑与几何复杂性的重要数学工具，对于理解包括聚合物和生物聚合物行为在内的细丝物理系统至关重要。琼斯多项式作为一种拓扑不变量，能够区分不同的纽结结构，这一特性对于表征蛋白质与DNA等分子链的构型具有基础性意义。传统的琼斯多项式计算方法受限于闭包方案和高昂的计算成本，对于现实世界中出现的复杂结构往往难以实用。本软件通过采用分治策略结合并行计算，并应用递归的Reidemeister变换来优化计算过程，将特定构型的计算时间复杂度从指数级降低至近似线性，从而显著缩短计算时间，使得在生物聚合物研究中实现更高效、更实用的应用成为可能。论文概述了软件的功能，以蛋白质结构为测试案例进行了详细的性能评估，并探讨了其对未来研究的启示以及潜在的算法改进方向。